I am working from Anton Deitmar's "Principles of Harmonic Analysis" and in one of the exercises he asks to show that the disk-algebra, equipped with the involution $$f^*(z) = \overline {f(\overline z)}$$ is a Banach-*-algebra. I have done the other part of the question but I am unsure how I would tackle this section.
The part I am stuck on is this:
$||f||=sup\{|f(z)|\ \Big|z\in\mathbb D\}$
$||f^*||=sup\{|f^*(z)|\ \Big|z\in\mathbb D\}=sup\{|\overline {f(\overline z)}|\ \Big|z\in\mathbb D\}$
So I guess I want to show that $sup\{|f(z)|\ \Big|z\in\mathbb D\}=sup\{|\overline {f(\overline z)}|\ \Big|z\in\mathbb D\}$ which I feel may be obvious but I cannot see why.
Just note that if $w=\overline{z}$, then $|f(z)|=|\overline{f(z)}|=|\overline{f(\overline{w})}|$. So the set of possible values of $|f(z)|$ is exactly the same as the set of possible values of $|\overline{f(\overline{z})}|$.